Consider a dynamical system given by the transformation T:[0,1)→[0,1) defined as: This is known as the *Bernoulli shift*. The space [0,1) with the Borel σ-algebra and the Lebesgue measure is our measure space. We have a sequence of random variables Xn defined by Xn(\omega) = T^n(\omega) \) for \( n \in \mathbb{N} \) and \( \omega \in [0,1) \). Using Birkhoff's Ergodic Theorem, compute the time average for the function f(x) = x with initial condition x_0 = 0.25 over 4 iterations.
Consider a dynamical system given by the transformation T:[0,1)→[0,1) defined as: This is known as the *Bernoulli shift*. The space [0,1) with the Borel σ-algebra and the Lebesgue measure is our measure space. We have a sequence of random variables Xn defined by Xn(\omega) = T^n(\omega) \) for \( n \in \mathbb{N} \) and \( \omega \in [0,1) \). Using Birkhoff's Ergodic Theorem, compute the time average for the function f(x) = x with initial condition x_0 = 0.25 over 4 iterations.
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter12: Quadratic Functions
Section12.8: Joint And Combined Variation
Problem 12MRE
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Consider a dynamical system given by the transformation T:[0,1)→[0,1) defined as:
This is known as the *Bernoulli shift*.
The space [0,1) with the Borel σ-algebra and the Lebesgue measure is our measure space.
We have a sequence of random variables Xn defined by Xn(\omega) = T^n(\omega) \) for \( n \in \mathbb{N} \) and \( \omega \in [0,1) \). Using Birkhoff's Ergodic Theorem, compute the time average for the function f(x) = x with initial condition x_0 = 0.25 over 4 iterations.
Transcribed Image Text:2x
2x1
if 0 < x < 0.5
if 0.5 x < 1
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